The Utility Indifference Price of the Defaultable Bond … Hu.pdf · Ito^ lemma, we can prove the...

Outline Introduction Problem Formulation Viscosity Solution Numerical Results The end

The Utility Indifference Price of the DefaultableBond in a Jump-Diffusion Model

Shuntai Hu Quan Yuan Baojun Bian

Department of MathematicsTongji University

2nd July, 2012

Shuntai Hu Indifference Price of the Defaultable Bond


1 Introduction

2 Problem Formulation

3 Viscosity SolutionValue Function as Viscosity SolutionComparison Principle

4 Numerical Results



Credit Risk

The global financial tsunami triggered by the U.S. subprimemortgage crisis drives the world economy into recession. Today theEuropean debt crisis is getting more and more deteriorative, whichmakes the investors have no confidence in the national credit ofsovereign state. One of the main reasons resulting in all of these isthat the participants in financial markets lack enough awareness forcredit risk.

Credit Risk

Credit risk, an important component consisting of financial risks, isan investors risk of loss arising from counterparties who cant orwould not like to fulfill their obligations on time.



Credit Risk Models

Structural Model Merton(1974), Black and Cox(1976)

Shortcoming

Credit spread verges to zero in the limit of zero maturity.

Intensity-based Model Ramaswamy and Sundaresan(1986),Litterman and Iben(1991), Jarrow andTurnbull(1995).

Feature

Characterize credit risk as an exogenous stochastic impact, thusavoiding the aforementioned defect.



Financial Derivative Pricing

No-arbitrage Valuation: Black and Scholes(1973)

Shortcomings

The framework is unable to capture and explain highpremiums observed in credit derivatives markets for unlikelyevents.

Most credit derivatives markets are OTC. There are manyconstraints on transaction. Some assets are illiquid or evennon-tradable.



Financial Derivative Pricing

Utility Indifference Valuation: Hodges and Neuberger(1984)

Features

Take the risk aversion of investors into account.

The methodology is dependent on the optimal portfolio theory.

The methodology is raised from utility indifference theory ineconomics.

Principle

Whether or not to hold the derivative which is treated as anotherinvestment instrument produce two optimal portfolio problems.The profit tendency of rationaleconomic men makes utilities of theabove two selections equivalent, and the derivative price is derivedfrom solving the utility equivalence equation.



Related Literature

Leung, Sircar and Zariphopoulou(2007): Indifference valuationfor defaultable bonds. Structural model of Black-Cox-type.GBM. The term-structure of the yield spread. Comparisonwith the Black-Cox model in the complete markets

Liang and Jiang(2009): Indifference valuation for defaultablebonds. Default happening at the maturity and at thefirst-passage-time. Recovery rate.

Sircar and Zariphopoulou(2006): Indifference valuation forcredit derivatives in single and two-name cases.Intensity-based model.



Assumptions

Two investible assets in the market: risk-free asset and riskyasset

No transaction cost

Financing and shorting are forbidden.

The investor has a utility function.



Investible Assets

Risk-free asset

bank account BtdBtBt

= rdt

Risky asset

stock StdStSt

= µdt+ σdWt +

∫ ∞−1

zN(dt, dz)

µ is the drift coefficient.

Wt is standard Brownian motion.

N(dt, dz) = N(dt, dz)− v(dz)dt, N(dt, dz) is randomPoisson measure, and v(dz) is the corresponding Levymeasure.



Wealth

Wealth process

The investor’s wealth Xt and the self-financing trading strategyπt ∈ [0, 1]

dXt = πtXtdStSt

+ (1− πt)XtdBtBt

Discount wealth process

Xt = e−rtXt, µ = µ− r

dXt = µπtXtdt+ σπtXtdWt +

∫ ∞−1

πtXtzN(dt, dz)



Defaultable Bond

At the maturity time T <∞, the investor gets 1 Dollar if nodefault occurs, or 0 if default occurs.

The default time τ is the first jump time of an exogenousPoisson process. For simplicity, we assume τ is an exponentialrandom variable with parameter β.

The investor could choose whether or not to hold the bond inhis investing period.



Two Opportunities for Investment

Not holding the defaultable bond

The value function

M(t, x) = supπ(·)∈A

E [U(XT )|Xt = x]

Holding the defaultable bond

The value function

V (t, x) = supπ(·)∈A

E [U(Xt+τ )I0≤τ≤T−t + U(XT + 1)Iτ≥T−t|Xt = x]

(1)

= supπ(·)∈A

Et,x

[∫ T

te−β(s−t)βU(Xs)ds+ e−β(T−t)U(XT + 1)

](2)



Dynamic Programming Equations

Not holding bond∂M

∂t+ supπ∈[0,1]

{LπM + BπM} = 0, (t, x) ∈ [0, T )× (0,+∞)

M(T, x) = U(x), x ∈ (0,+∞)

Holding bond∂V

∂t+ supπ∈[0,1]

{LπV + BπV } − βV + βU = 0,

(t, x) ∈ [0, T )× (0,+∞)

M(T, x) = U(x+ 1), x ∈ (0,+∞)

(3)

Lπϕ =

(µ−

∫ ∞−1

zv(dz)

)πx∂ϕ

∂x+σ2

2π2x2

∂2ϕ

∂x2

Bπϕ =

∫ ∞−1

[ϕ(t, x(1 + πz)

)− V (t, x)

]v(dz)



The Bond Price

If we can obtain the value functions M,V , then the utilityindifference price p0 of the defaultable bond is a solution of thefollowing equation

M(0, x) = V (0, x− p0)


Outline Introduction Problem Formulation Viscosity Solution Numerical Results The endValue Function as Viscosity Solution Comparison Principle

Outline

1 Introduction



4 Numerical Results



Viscosity Solution

Consider the general nonlinear integro-differential parabolicequation

−∂u∂t

+ F (t, x, u,Bu,Du,D2u) = 0, (t, x) ∈ OT = (0, T )×O,

where

Bu =

∫RN

[u(t, x+ z)− u(t, x)] dpt,x(z).

In addition, an elliptical condition is assumed for F , u isquasi-monotone and decreasing in the nonlocal item Bu.



Viscosity Solution

Definition

u(t, x) is the viscosity subsolution(supersolution) of the aboveequation if u(t, x) is upper(lower)-semicontinuous and one of thefollowing equivalent items holds:

−q + F (t, x, u(t, x),Bu, p,X) ≤ 0(≥ 0), for∀(q, p,X) ∈ P 2,+

O (t, x)(P 2,−O (t, x)), where P 2,+

O , P 2,−O are the

parabolic semijets.

−∂φ∂t (t, x) + F

(t, x, u(t, x,Bφ,Dφ(t, x), D2φ(t, x)

)≤ 0(≥ 0),

for each smooth function φ such that u− φ has a localmaximum(resp., a minimum) at (t, x).

−∂φ∂t (t, x) + F

(t, x, u(t, x,Bφ,Dφ(t, x), D2φ(t, x)

)≤ 0(≥ 0),

for each smooth function φ such that u− φ has a global strictmaximum(resp., a minimum) at (t, x).



Value Function is Viscosity Solution

We mainly consider the equation (3) in the case of holding thedefaultable bond. Using the dynamic programming principle andIto lemma, we can prove the following theorem.

Theorem

Assume the value function defined by (1) is continuous, and thenit is a viscosity solution of the HJB equation (3).

The dynamic programming principle is as follows

V (t, x) = supπ(·)∈A

E

[∫ t+h

te−β(s−t)βU(Xs)ds+ e−βhV (t+ h,Xt+h)

],

for ∀0 < h < T − t.



Outline

1 Introduction



4 Numerical Results



Change Variable

For simplicity, we consider the Levy process as jump-diffusionprocess, i.e. v(dz) = λp(z)dz, where p(z) is the density functionof jump amplitude which satisfies

∫∞−1 zp(z)dz = κ. The utility

function U(x) = −e−γx, thus the value function is bounded.Let us make a change of variabley = lnx ∈ (−∞,+∞),W (t, y) = V (t, x), and now the HJBequation (3) becomes

∂W

∂t+H

(∂W

∂y,∂2W

∂y2,W

)− βW + βU(ey) = 0,

W (T, y) = U(ey + 1), y ∈ (−∞,+∞)

(4)

H(p,A,W ) = supπ∈[0,1]

{[(µ− λκ)π − σ2

2π2]p+

σ2

2π2A

+ λ

∫ ∞−1

[W (t, y + ln(1 + πz))−W (t, y)]p(z)dz

}Shuntai Hu Indifference Price of the Defaultable Bond


Comparison Principle

For equation (4), we can prove the following comparison principle.

Theorem

For density function of jump amplitude, we assumesupπ∈[0,1]

∫−1∞|ln(1 + πz)|2 p(z)dz <∞. If u, v are respectively

bounded viscosity subsolution and viscosity supersolution ofequation (3) such that u(T, y) ≤ v(T, y), then in [0, T )× R wealso have u(t, y) ≤ v(t, y).

Using the comparison principle, we can show that the valuefunction is the unique viscosity solution for equation (3).



Setup

The density function of jump amplitude of stock price

p(z) =1√

2π(1 + z)exp

{− ln2(1 + z)

2

}i.e. ln(1 + Z) has distribution N(0, z).

The utility function: U(x) = −e−γx

Parameters:

drift µ = 0.07volatility σ = 0.3intensity of Poisson process in jump-diffusion λ = 0.1risk aversion coefficient γ = 1default intensity β = 0.03



Value Function

The figure of the value function of not holding the defaultablebond:



Bond Price vs Risk Aversion

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Risk Aversion Coefficient γ

Bon

d P

rice

Bond price isdecreasing inthe risk aversioncoefficient.Intuitively, themore theinvestor aversethe credit risk,the lower thebond’s value isfor the investor.



Bond Price vs Default Intensity

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.110.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

Default Intensity β

Bon

d P

rice

Bond price isdecreasing in thedefault intensity.Intuitively, thelarger defaultintensitysuggests higherdefault risk.



Bond Price vs Volatility

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9505

0.9505

0.9505

0.9505

0.9505

0.9505

0.9505

0.9505

0.9505

0.9505

Volatility σ

Bon

d P

rice

Bond price is increasingin the volatility ofGBM. The larger σ,reflecting higher risk ofthe stock, makes theattraction of stockpoorer to investors. Inthe samecircumstances,investors prefer thebond to the stock, thusleading to a higherprice of the bond.



The end


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